Micro Heterogeneity and Aggregate Consumption Dynamics

Transcription

1 Micro Heterogeneity and Aggregate Consumption Dynamics SeHyoun Ahn Greg Kaplan Benjamin Moll Thomas Winberry Christian Wolf January 31, 217 Abstract [TO BE RE-WRITTEN] We study the aggregate consumption, interest rate and output dynamics of a heterogeneous agent economy that is parameterized to match key features of the cross-sectional distribution of labor income, wealth, and marginal propensities to consume measured from household-level micro data. Households face a process for idiosyncratic income risk with leptokurtic growth rates and can self-insure in two assets with different degrees of liquidity. The equilibrium features a threedimensional distribution that moves stochastically over time, rendering computation difficult with existing methods. We develop computational tools to efficiently solve a broad class of heterogeneous agent model with aggregate shocks that include our model as a special case. The method uses linearization to solve for the dynamics of a reduced version of the model, which is obtained from a model-free dimensionality reduction method for the endogenous distributions. We will publish an open source set of Matlab codes to implement our method in an easy-to-use and model-free way. We find that our model, which is parameterized to household level facts, is consistent with the sensitivity of aggregate consumption to predictable changes in aggregate income, and with the relative smoothness of aggregate consumption - features that are difficult to generate in representative agent. We illustrate the usefulness of our model and methods for studying the distributional implications of shocks more generally. Ahn: PrincetonUniversity, sehyouna@princeton.edu,Kaplan: UniversityofChicagoandNBER, gkaplan@uchicago.edu; Moll: Princeton University, CEPR and NBER, moll@princeton.edu; Winberry: University of Chicago, Thomas.Winberry@chicagobooth.edu; Wolf: Princeton University, christian.k.wolf@gmail.com. We thank Chris Carroll, Chris Sims, Jonathan Parker and Stephen Terry for useful comments. 1

2 1 Introduction [PLEASE NOTE: INTRODUCTION WILL BE RE-WRITTEN FOR CONFERENCE VER- SION] Over the last twenty years, tremendous progress has been made in developing models consistent with the rich heterogeneity in income, wealth, and consumption behavior across households in the micro data. These models often have strikingly different implications about monetary and fiscal policies than representative agent models and allow us to study the policies distributional implications across households. 1 However, these policy implications are only relevant to the extent that the models are consistent with the time-series behavior of consumption in the aggregate data. This is still largely an open question due to the many computational difficulties in solving and analyzing heterogeneous agent macro models. We make two main contributions in this paper. First, we develop an efficient and easyto-use computational method for solving a wide class of dynamic heterogeneous agent macro models. To make the methodas accessible aspossible, we will publish an opensource suite of Matlab codes which implement the method in an easy-to-use and model-free way. Second, we use the method to analyze the time series behavior of a heterogeneous agent model parameterized to match the distribution of income, wealth, and marginal propensities to consume in the micro data. Building on Kaplan, Moll and Violante (216), our model features leptokurtic income shocks and two assets with different degrees of liquidity. We find that the model is jointly consistent with two key facts about aggregate dynamics: consumption responds to predictable changes in income and is substantially less volatile than income. Matching these two facts together has proven a challenge for traditional spender-saver models, suggesting our two-asset model is a promising framework for studying aggregate consumption dynamics going forward. Our computational methodolofy extends standard linearization techniques, routinely used to solve representative agent models, to the heterogeneous agent context. 2 We first solve for the nonstochastic steady state of the model using a globally accurate nonlinear approximation. The approximation gives a discretized representation of the model s stationary equilibrium, including a non-degenerate distribution of agents over their individual state variables. We then compute a first-order Taylor expansion of the dynamic version of the discretized model around the nonstochastic steady state. This results in a large, but linear, 1 For examples studying fiscal policy, see McKay and Reis (213) and Kaplan and Violante (214); for monetary policy, see McKay, Nakamura and Steinsson (215), Auclert (214), and Kaplan, Moll and Violante (216). 2 As we discuss in more detail below, the use of linearization to solve heterogeneous agent economies is not new. Our method builds on the ideas of Dotsey, King and Wolman (1999), Campbell (1998), and Reiter (29), and is related to Preston and Roca (27). 2

3 system of equations, which we solve using standard solution techniques. Although our solution relies on linearizing this system, it is still nonlinear in individual state variables, which preserves potentially nonlinear aggregate dynamics implied by movements in the distribution of agents. 3 Our two-asset consumption model is so large that the size of the linear system is numerically intractable. We develop a model-free reduction method to reduce its size. Our method generalizes Krusell and Smith (1998) s insight that only a small subset of the information contained in the distribution of agents is necessary to accurately forecast the variables needed to solve the model. Krusell and Smith (1998) s procedure posits a set of moments that capture this information based on economic intuition, and verifies its accuracy ex-post using a forecast-error metric; our method instead leverages advances in engineering to allow the computer the identify the necessary information in a completely model-free way. Households in our model save in a liquid and illiquid asset. The illiquid asset earns a high return but adjusting it requires a transaction cost, while the liquid asset earns a low return but adjusting it is less costly. A substantial portion of households endogenously choose to save all their wealth in illiquid assets, earning the higher return but not paying the transaction cost to smooth consumption in response to small shocks. These hand to mouth consumers have high marginal propensities to consume (MPCs), in line with empirical evidence presented in Johnson, Parker and Souleles (26), Parker et al. (213) and Fagereng, Holm and Natvik (216) among others. The presence of hand to mouth households allows the model to match both the sensitivity of aggregate consumption to predictable changes in aggregate income as well as its smoothness relative to unconditional income fluctuations. Over short horizons, these high MPC households are sensitive to changes in income; however, over long horizons they adjust their illiquid asset holdings in order to smooth their consumption profiles, generating smoothness. Jointly matching sensitivity and smoothness has posed a challenge for many benchmark models in the literature. Simple spender-saver models directly assume that an exogeneous fraction of households are permanently hand to mouth; these models generate sensitivity by construction, but because hand to mouth households never smooth consumption they overstate volatility. One-asset incomplete markets models, in the spirit of Aiyagari (1994) and Krusell and Smith (1998), endogenize the amount of hand to mouth households with a simple borrowing constraint. However, standard calibrations of these models feature far too few high-mpc households relative to the data to generate sensitivity; we find, for example, 3 However, since our method does deliver a linear state-space representation of the equilibrium, it can also be used for likelihood-based estimation. 3

4 that the consumption response to a shock in the one-asset version of our model is nearly 4% smaller than in the two-asset model. We therefore conclude that our two-asset model provides a promising framework for studying aggregate consumption and the policies meant to affect it going forward. [TO BE ADDED] Our model is particularly useful for studying the distributional consequences of aggregate shocks, and for analyzing the aggregate effects of shocks that differentially impact households at different parts of the income, asset and consumption distributions. In the final part of the paper we illustrate these effects through a series of experiments that cannot be examined in a representative agent framework. Related Literature [TO BE COMPLETED] Our paper relates to two main strands of literature. On the computational side, we build on Achdou et al. (215), who develop continuous-time tools to solve heterogeneous agent models without aggregate shocks. We add aggregate shocks using a mix of globally and locally accurate approximations, similar to Dotsey, King and Wolman (1999), Campbell (1998), and Reiter (29). Our model-free reduction method builds on Amsallem and Farhat (211) and Antoulas (25) in the engineering literature. Also related is Preston and Roca (27) who linearize with respect to both aggregate and individual state variables, which requires that decisions are a smooth functions of individual state variables as well. We do not pursue this strategy because in our model there are numerous kinks in individual behavior and idiosyncratic shocks are large. On the consumption dynamics side, our two-asset model structure comes from Kaplan and Violante (214) and Kaplan, Moll and Violante (216). The aggregate facts we focus on are drawn from Campbell and Mankiw (1989). Road Map Before studying the dynamics of the two-asset model model, Section 2 describes the details of our computational method. We then apply the method to analyze the two-asset model in Section 3. 2 Computational Method We introduce our method in the context of the Krusell and Smith (1998) model. This model provides a natural expository tool because it is well-known and substantially simpler than the two-asset model in Section 3. As we show in Section 3, the method is applicable to a much broader class of models. 4

5 Continuous Time Our method requires the model to be specified in continuous time. While discrete time poses no conceptual difficulty (in fact, Dotsey, King and Wolman(1999), Campbell (1998), and Reiter (29) originally proposed this general approach in discrete time), working in continuous time has three key numerical advantages that we heavily exploit: 1. The matrices characterizing the model s equilibrium conditions are naturally sparse in continuous time; intuitively, agents only drift an infinitesimal amount in the state space in an infinitesimal unit of time. In large models such as in Section 3, these matrices are extremely high-dimensional and sparsity is necessary to even store them. 2. Continuous time allows us to handle non-convexities in individual decision problems, such as fixed costs, much more easily than in discrete time. Non-convexities are necessary to match micro data in many heterogeneous agent models, such as with menu costs in firms pricing decisions, adjustment costs in firms investment decisions, or transaction costs in households portfolio allocation decisions. 3. Continuous time allows us to more easily capture the nonlinear relationship between aggregate variables and the distribution of agents in the economy, despite using linear approximation methods. As we discuss in Section 2.6, the restriction imposed by linearity is most severe upon impact of the shock, which is arbitrarily small in continuous time. 2.1 Krusell-Smith Model Description Environment There is a fixed mass of households j [,1] who have preferences represented by the expected utility function E c1 γ ρt jt e 1 γ dt, where ρ is the rate of time preference and γ is the coefficient of relative risk aversion. At each instant t, a household s idiosyncratic labor productivity is z jt {z L,z H } with z L < z H. Households switch between the two values for labor productivity according to a Poisson process with arrival rates λ L and λ H. 4 A household with labor productivity z jt earns labor income w t z jt. Markets are incomplete; households can only trade in productive capital a jt subject to the borrowing constraint a jt a. 4 The assumption that idiosyncratic shocks follow a Poisson process is for simplicity of exposition; the method can also handle diffusion or jump-diffusion shock processes. 5

6 There is a representative firm which has access to the Cobb-Douglas production function Y t = e Zt K α t N 1 α t, where Z t is aggregate productivity, K t is aggregate capital and N t is aggregate efficiency units of labor. Aggregate productivity follows the Ornstein-Uhlenbeck process dz t = νz t dt+σdw t, where dw t is the innovation to a standard Brownian motion, ν captures the rate of mean reversion, and σ captures the size of innovations. Equilibrium The household-level state variables are capital holdings a and the idiosyncratic labor productivity shock z. The aggregate state variables are the aggregate productivity shock Z t and the cross-sectional distribution of households over their individual states g t (a,z). For notational convenience we denote an equilibrium object conditional on a particular realization of the aggregate state (g t (a,z),z t ) with a subscript t. An equilibrium of the model is then characterized by the following equations: ρv t (a,z) = max c u(c)+ a v t (a,z)(w t z +r t a c) +λ z (v t (a,z ) v t (a,z))+ 1 dt E t[dv t (a,z)] (HJB) t g t (a,z) = a [s t (a,z)g t (a,z)] λ z g t (a,z)+λ z g t (a,z ) (KFE) dz t = νz t dt+σdw t (TFP) w t = (1 α)e Zt Kt α N α, (AUX 1) r t = αe Zt Kt α 1 N 1 α δ, (AUX 2) K t = ag t (a,z)dadz. (AUX 3) and where s t (a,z) is the optimal saving policy function corresponding to the household optimization problem (HJB). For detailed derivations of these equations, see Achdou et al. (215). (HJB) is the household s Hamilton-Jacobi-Bellman equation; due to our use of timedependent notation with respect to aggregate states, E t denotes the conditional expectation with respect to aggregate states only. (KFE) is the Kolmogorov Forward Equation and describes how the mass of households at a point in the individual state space evolves over time. (TFP) describes the evolution of the aggregate shock. Lastly, (AUX 1) to (AUX 3) define prices given the aggregate state. 6

7 2.2 Computational Method Our method consists of three broad steps. First, we solve for the non-stochastic steady state of the model without aggregate shocks. Second, we take a first-order Taylor expansion around the steady state, yielding a system of stochastic linear differential equations. Third, we solve that linear system using standard techniques. Conceptually, each of these steps is a straightforward extension of linearization for representative agent models to the heterogeneous agent context. However, the size of heterogeneous agent models leads to a number of computational challenges. Step 1: Compute the Steady State In a heterogeneous agent model, the nonstochastic steady state features idiosyncratic shocks at the household level, which implies that households decision rules are a function of individual state variables v(a,z) and that there is a non-degenerate stationary distribution of households g(a, z). We use a non-linear approximation of these functions in order to retain the rich non-linearities and heterogeneity in behavior at the individual level. In principle, any method to compute the steady state can be used in this step; we use the finite difference methods outlined in Achdou et al. (215) because they are fast, accurate, and robust. The finite difference method approximates the value function and distribution over a discretized grid of asset holdings a =(a 1 = a,a 2,...,a I ) T. Denote the value function and distribution along this discrete grid using the vectors v = (v(a 1,z L ),...,v(a I,z H )) T and g = (g(a 1,z L ),...,g(a I,z H )) T. Both v and g are of dimension N 1 where N = 2I is the total number of grid points in the state space. We solve the steady state versions of (HJB) and (KFE) at each point on this grid, approximating the partial derivatives using finite differences. Achdou et al. (215) show that if the finite difference approximation is chosen correctly, the discretized steady state can be written compactly as the following system of matrix equations: ρv = u(v)+a(v;p)v (HJB SS) = A(v;p) T g (KFE SS) p = F(g). (PRICE SS) (HJB SS) is the approximated (HJB) for each point on the discretized grid, expressed in our vector notation. The vector u(v) is the maximized utility function over the grid and the matrix multiplication A(v; p) v captures the remaining terms in (HJB). These terms are differences of the value function at different points in the grid, weighted by how fast 7

8 households move in the individual state space. (KFE SS) is the discretized version of (KFE) in steady state and imposes that the distribution is stationary. Finally, (PRICE SS) defines the prices p = (r,w) T as a function of aggregate capital through the distribution g. Since v and g each have N entries, the total system has 2N + 2 equations in 2N + 2 unknowns. In simple models like this one, highly accurate solutions can be obtained with as little as N = 1 grid points (i.e., I = 5 asset grid points together with the two income states); however, in more general models with complex shock processes and multiple individual state variables, such as the two-asset model in Section 3, N can easily grow into the tens of thousands. Exploiting the sparsity of the transition matrix A(v; p) is necessary to even numerically represent the steady state of these large models. Step 2: Compute First-Order Taylor Expansion The second step of our method is to compute a first-order Taylor expansion of the model s discretized equilibrium conditions around the nonstochastic steady state. With aggregate shocks, the discretized equilibrium takes the form ρv t = u(v t )+A(v t ;p t )v t + 1 dt E tdv t dg t dt = A(v t;p t ) T g t p t = F(g t ;Z t ) (NON-LINEAR) dz t = νz t dt+σdw t. With shocks to TFP Z t, the aggregate state (g t,z t ) fluctuates over time, inducing fluctuations in marginal products and therefore in prices p t = F(g t ;Z t ). Fluctuations in prices in turn induce fluctuations in households decisions and therefore in v t and the transition matrix A(v t ;p t ). We compute the first-order derivatives of the system using automatic differentiation, which is a computational technique which is faster than symbolic or numerical differentiation and is accurate up to machine precision. 5 The main challenge in computing the derivatives is the size of the system again, 2N + 2 equations with 2N + 2 variables. Exploiting the sparsity of the transition matrix A(v t ;p t ) is essential for numerical feasibility, especially in large models. Unfortunately, to the best of our knowledge, there is no open-source automatic 5 Automatic differentiation exploits the fact that the computer represents any function as the composition of various elementary functions, such as addition, multiplication, or exponentiation, which have known derivatives. Automatic differentiation builds the derivative of the original function by iteratively applying the chain rule. 8

9 differentiation package for Matlab which exploits sparsity. We therefore wrote our own package which will be made publicly available in the Matlab code suite. The first-order Taylor expansion of (NON-LINEAR) can be written as: 6 E t d v t B vv B vp v t dĝ t = B gv B gg B gp ĝ t B pg I B pz p t dt dz t } {{ ν Z t } B (LINEAR) The variables in the system v t, ĝ t, p t, and Z t are expressed as deviations from their steady state values, and the matrix B is composed of the derivatives of the equilibrium conditions evaluated at steady state. Since the pricing equations are static, the third row of this matrix equation only has non-zero entries on the right hand side. 7 Step 3: Solve Linear System The final step of our method is to solve the linear system of stochastic differential equations (LINEAR). Following standard practice, we perform a Schur decomposition of the matrix B to identify the stable and unstable roots of the system. If the Blanchard and Kahn (198) condition holds, i.e., the number of stable roots equals 6 To arrive at (LINEAR), we first rearrange (NON-LINEAR) so that all time derivatives are on the lefthand side. Second, we take the expectation of the entire system (not just the first equation) using the fact that the expectation of a Brownian increment is zero E t [dw t ] =, to write (NON-LINEAR) compactly without the stochastic term as E t dv t dg t = dz t u(v t ;p t )+A(v t ;p t )v t ρv t A(v t ;p t ) g t F(g t ;Z t ) p t νz t dt. Finally, we linearize this system to arrive at (LINEAR). In general, this loses information contained in the stochastic term dw t ; however, at first order this is without loss of generality because certainty equivalence holds. 7 The fact that we can express prices as a static function of ĝ t and Z t is a special feature of the Krusell- Smith model. More generally, the market clearing conditions involve integrals of decisions against the distribution and only define prices implicitly. In those cases, one must approximate the time derivative of the market clearing condition (rather than the static market clearing condition) in order for the resulting the linear system to be of full rank. 9

10 the number of state variables, then we can compute the solution 8 v t = D vg ĝ t +D vz Z t p t = D pg ĝ t +D pz Z t. (SOLUTION) 2.3 Model-Free Reduction Method Solving the linear system (LINEAR) is extremely fast using the approach above because the Krusell-Smith model is relatively small. However in larger models, such as the two-asset model in Section 3, the required matrix decomposition becomes prohibitively expensive. In order to solve these more general models we must therefore reduce the total size of the system. In this subsection, we describe a set of model reduction techniques from the engineering literature to reduce the total size of the system without sacrificing numerical accuracy. Sections and describe how to reduce the distribution g t and Section describes how to reduce the value function v t An Economist-Friendly Introduction to State Space Reduction We reduce the distribution g t using a set of tools known as state space reduction. In this section, we will provide a brief introduction to these tools; in the next subsection, we will apply them to the Krusell-Smith model. The following discussion is based on Amsallem and Farhat (211), which in turn builds on Antoulas (25, Section 1.1.1). 1 In order to make the material accessible to economists, we exploit a strong analogy with leastsquares regression. For the ease of exposition, in this subsection we make three key simplifications to our problem (LINEAR). First, we assume that there is no aggregate uncertainty σ = so that we only analyze deterministic transition paths starting away from steady state. 11 Second, we 8 We have written the price vector p t as a linear function of the state vector to easily exposit our methodology in a way that directly extends to more general models. However, this approach is not necessary in the Krusell-Smith model because we can simply substitute the expression for prices directly into the households budget constraint. 9 Reiter (21) discusses related state space reduction techniques based on the observability matrix that we describe below. However, he pursues an approach based on singular value decompositions, which are numerically unstable for large models such the one in Section 3. 1 All lecture notes for Amsallem and Farhat (211) are available online at and the entire book by Antoulas (25) is available at 11 Because certainty equivalence holds in a linearized economy, these transition paths characterize the impulse response dynamics of the model after impact of a shock. 1

11 assume thatthepricevector p t isascalar andwritep t ; thegeneralizationtoanl-dimensional price vector is straightforward and discussed below. Third, we assume B gv = B gp =, which implies that the evolution of the distribution and prices can be characterized separately from the evolution of the value function v t : dg t dt = B ggg t p t = b pg g t. (SIMPLE) Here b pg B pg is an 1 N vector. This simplified formulation of the problem focuses on the role of the distribution in determining the current price and, through the evolution of the distribution over time, the path of future prices. In subsection we will return to the full model using the insights developed from analyzing (SIMPLE). The insight we exploit is that only a small subset of the information in g t is necessary to accurately compute the path of prices p t. In fact, in the discrete time version of this model, Krusell and Smith (1998) show that just the mean of g t is sufficient according to a forecast-error metric. However, because it relies on the economic properties of the model, it is not obvious how to generalize Krusell and Smith (1998) s approach to other environments. The state space reduction technique we pursue generalizes their approach in a model-free way, allowing the computer to compute the moments of the distribution necessary to forecast prices. We say that the model (SIMPLE) exactly reduces if there exists a k S -dimensional subspace S with k S << N such that g t = β 1t x 1 +β 2t x β ks tx ks, where X S = [x 1,...,x ks ] R N k S is a basis for the subspace S and β 1t,...,β ks t are scalars. If we knew the time-invariant basis X S, this would decrease the dimensionality of the problem because the distribution would be characterized by the k S dimensional vector of coefficients β t. Typically exact reduction as described above will not hold, so we instead must estimate a trial basis X = [x 1,...,x k ] R N k such that the model approximately reduces, i.e., g t β 1t x 1 +β 2t x β kt x k or, in matrix form, g t Xβ t. Because this relationship only holds approximately, it is 11

12 convenient to write it as g t = Xβ t +ε t, (1) where ε t R N is a residual. The formulation (1) is a standard linear regression in which the distribution g t is the dependent variable, the basis vectors X are the dependent variables, and the coefficients β t are to be estimated. For now, suppose that we know the trial basis X; in this case we can estimate the coefficients by imposing the orthogonality condition X T ε t =, which gives the familiar estimator β t = (X T X) 1 X T g t. (2) Any sensible trial basis will be orthonormal, so that (X T X) 1 = I, further simplifying (1) to β t = X T g t. We can compute the evolution of this coefficient vector by differentiating (2) with respect to time to get dβ t dt = d dt XT g t = X T B gg Xβ t. (3) Putting (2) and (3) together, we have the reduced system dβ t dt = XT B gg Xβ t p t = b pg Xβ t. (REDUCED) Summing up, assuming we have a good trial basis X, this procedure takes us from a system of N differential equations for g t in (SIMPLE) to a system of k << N differential equations for β t in (REDUCED). Choosing the Regressors X The success of this model reduction strategy relies crucially on the choice of the trial basis X. Again, we choose the trial basis with our goal of computing the path of prices p t as accurately as possible. In continuous time, it is natural to operationalize the notion accurately matching the path of prices p t by matching the Taylor series approximation of p t+s around p t : p t+s p t +ṗ t s+ p t s p (k 1) t s k 1, (4) where ṗ t, p t, and p (k 1) t denote time derivatives. From (SIMPLE) we can write these time 12

13 derivatives as or, in vector form, P(t) := p ṗ p. p (k 1) ṗ = b pg ġ = b pg B gg g p = b pg B 2 gg g p (k 1) = b pg B k 1 gg g, b pg B gg = O(b pg,b gg )g t, where O(b pg,b gg ) := b pg B 2. (5) gg. b pg Bgg k 1 O(b pg,b gg ) is known as the observability matrix. 12 Using this notation, we can write the kth order Taylor expansion of p t+s as p t+s [1,s,s 2,...,s k 1 ]P(t). b pg In general, p t+s depends on the current distribution g t ; setting the trial basis X = O(b pg,b gg ) T allows us to exactly match the matrix P(t) and, therefore, the kth-order Taylor expansion of p t+s given knowledge of the reduced vector β t only. To see this, substitute (1) into (5) to get P(t) = O(b pg,b gg )Xβ t +O(b pg,b gg )ε t, where againε t is the residual fromthe regression. By construction, the residuals must satisfy X T ε t = O(b pg,b gg )ε t =, implying that P(t) = O(b pg,b gg )Xβ t exactly. Appendix A.1 establishes an even stronger result: this choice of trial basis matches the entire Laplace Transform of the transition path (SIMPLE). 12 Observability of a dynamical system is an important concept in control theory introduced by Rudolf Kalman, the inventor of the Kalman filter which is widely used in economics. It is a measure for how well a system s states (here g t ) can be inferred from knowledge of its outputs (here p t ). For linear time-invariant (LTI) systems it can be directly inferred from the observability matrix O(b pg,b gg ). 13

14 Practical Considerations Although conceptually straightforward, the choice of trial basis X = O(b pg,b gg ) T is numerically unstable due to approximate multicollinearity. As in standard regression, high degree standard polynomials are nearly collinear due to the fact that, for large k, B k 2 gg B k 1 gg, leaving the necessary regression of the distribution onto X numerically intractable. We overcome this challenge using Krylov subspace methods, an equivalent but more numerically stable class of methods. For any N N matrix A and N 1 vector b, the order-k Krylov subspace is K k (A,b) = span ({ b,ab,a 2 b,...,a k 1 b }) It turns out that the regression of g t on O(b pg,b gg ) T is equivalent to the projection of g t onto the the order-k Krylov subspace generated by B T gg and bt pg, K k(b T gg,bt pg ). There are many methods to compute this projection in the literature; we have found that Arnoldi iteration is a fast and robust procedure. 13 Higher-Dimensional price vector p t R l. Generalizing our model reduction strategy to an l-dimensional price vector p t is straightforward. In this case the second equation in (SIMPLE) isreplaced by p t = B pg g t where B pg is l N. Thelogicforchoosing thetrial basis remains unchanged. We therefore choose X = O(B pg,b gg ) which is now lk N, implying that there are lk independent variables in the regression (1) instead of k. Our discussion of model reduction has not relied on the fact that the vector p t literally consists of prices; it is simply the vector of objects we wish to accurately describe. In practice, we often also include other variables of interest, such as aggregate consumption C t or output Y t, to ensure the reduced model accurately describes their dynamics as well Reducing the Distribution in the Full Model There are two main complications in the full model (LINEAR) relative to the simpler model (SIMPLE) we just analyzed. First, we must additionally keep track of the value function v t 13 The literature also presents alternatives to our least-squares approach to computing the coefficients β t. In particular, one can also estimate β t using what amounts to an instrumental variables strategy: one can define a second subspace spanned by the columns of some matrix Z and impose the orthogonality condition Z T ε t =. This yields an alternative estimate β t = (Z T X) 1 Z T g t. Mathematically, this is called an oblique projection (as opposed to an orthogonal projection) of g t onto the k-dimensional subspace spanned by the columns X along the kernel of Z T. See Amsallem and Farhat (211) and Antoulas (25) for more detail on oblique projections. 14

15 and the aggregate TFP shock Z t, both of which we previously ignored by setting B gv = and assuming away aggregate uncertainty. The need to keep track of v t and Z t results in a larger system. Second, there is a feedback from prices p t to the distribution g t which we previously ignored by setting B gp = and a similar feedback from prices to the value function v t. To deal with this issue, solve the third line of (LINEAR) to get p t = B pg ĝ t +B pz Z t. (6) Substituting this expression for prices p t back into (LINEAR), we arrive at E t d v t B vv B vp B pg B vp B pz v t dĝ t = B gv B gg +B gp B pg B gp B pz ĝ t dt (7) dz t ν Now that we have eliminated prices, the total system has a similar form as (SIMPLE) except that there the additional variables v t and Z t. This is a system of 2N linear ordinary differential equations, i.e. the dimension of the dynamical system is 2N, where again N is the total size of the grid. 14 Although our full system (7) is larger than (SIMPLE), we can follow the same steps as in Section Assuming we have an orthonormal trial basis X, run the same regression (1) as before to get the reduced coefficient vector β t = X T ĝ t. Differentiate this expression with respect to time to get the evolution of the coefficient vector dβ t dt = [ X T B gv v t +X T (B gg +B gp B pg )Xβ t +X T B gp B pz Z t ], (8) which is of the same general form as before, with more complicated component matrices since we substituted the price vector into the system directly. We again choose our trial basis X to replicate the Taylor series expansion of E t [ p t+s ] around p t. In computing this Taylor expansion we make the assumption that B gv = in (7). This assumption is only to make the choice of trial basis transparent; we do not use it elsewhere to solve the model. To proceed, it is convenient to define the expanded state vector ŷ t = ] [ĝt 14 In this calculation, we drop one redundant equation in the law of motion for the distribution that results from the fact that the distribution must integrate to 1. Z t Z t 15

16 and to write (6) and (7) as p t = B py ŷ t (9) [ ] [ ] d v t B vv B vp B py ][ vt E t = dt (1) dŷ t B yv B yy +B yp B py where the matrices B yv,b yy,b yp and B py are found from (LINEAR) and given by [ ] B gv B yv :=, B yy := [ B gg ν ], B yp := [ B gp ] ŷ t, B py := From (9) and (1), and our temporary assumption B gv, we then have [B pg B pz ]. E t [d p t ] B py (B yy +B yp B py )ŷ t dt E t [d 2 p t ] B py (B yy +B gx B py ) 2 ŷ t dt 2 and so on. Following the same steps as 2.3.1, the observability matrix of this system has the same structure as (5) but with b pg B gg replaced by B py (B yy + B yp B py ). As before, we set X = O T Reducing the Value Function After reducing the dimensionality of the distribution ĝ t, we are left with a system of dimension N + k with k << N. Although this is considerably smaller than the original system which was of size 2N, it is still large because it contains N equations for the value function at each point along the individual state space. In complex models, this leaves the linear system too large for matrix decomposition methods to be feasible. 16 We use a spline approximation to reduce the dimensionality of the steady-state value function v t. In most models, the value function is sufficiently smooth that a low-dimensional spline provides an accurate approximation. Any spline approximation can be represented by 15 A key practical consideration when performing the model reduction is that, even though B yy is sparse and B yp and B py are only l N + 1, the matrix B yy + B yp B py which actually enters the system (7) is N+1 N+1 and not sparse. In our application, N = 66,, and even storing this matrix in Matlab is not feasible. Fortunately it is never actually necessary to compute this full matrix; instead, it is only necessary to compute B py (B yy +B yp B py ), which is the sum of two l N +1 matrices: B py B yy and (B py B yp )B py. 16 One way to overcome this challenge is to use sparse matrix methods to find just the k eigenvalues associated with the stable eigenvectors. This is much faster than computing the full matrix decomposition necessary to obtain the full set of eigenvectors. However, it is slower than the approach we pursue in this subsection. 16

17 a linear change of basis from the original value function v t to spline knot points ω t v t Sω t where S is a N k v matrix defining the k v -dimensional subspace in which the approximation lives andk v isthenumber ofknot points. Weapproximate thedeviationofthevaluefunction from the steady-state with a quadratic spline, rather than the value function itself. We have found that non-uniformly spaced quadratic splines work extremely well for four reasons. First, the quadratic spline exploits the smoothness of the underlying value function to substantially reduce dimensionality. Second, the non-uniform spacing can be used to place knots in regions of the state space with curvature, allowing for an efficient dimensionality reduction. Third, the quadratic spline preserves monotonicity between knot points, which is important in computing first-order conditions. Fourth, the local nature of quadratic splines does not create spurious oscillations at boundary conditions, which often occurs with global approximations. Our Matlab code suite provides codes to implement the non-uniformly spaced quadratic spline. It is important to note the difference between approximating the steady-state value functions using quadratic splines after having solved for these functions using finite difference methods (which we do), versus solving for the steady-state value functions in the space of quadratic splines (which we do not do). The finite difference method we use to solve the steady-state does not impose that the value function is everywhere differentiable. This is potentially important for capturing the effects of non-convexities such as fixed costs and discrete choices. However, after having solved for the steady-state value functions, it is typically the case that these functions have kinks at only a finite number of points, and are well approximated by smooth functions between these points. It is then straightforward to fit quadratic splines between the points of non-differentiability Putting Things Together: the Reduced Linear System Summarizing the previous Sections and 2.3.3, we have approximated the value function and distribution as v t Sω t and ĝ t Xβ t which in turn implies that the coefficient vectors are given by ω t = (S T S) 1 S T v t = S T v t β t = (X T X) 1 X T ĝ t = X T ĝ t (11) 17

18 where we have used that both S and X are orthonormal bases. Now we simply need to keep track of the k v 1 coefficient vector ω t for the value function and the k g 1 coefficient vector for the distribution. Given knowledge of these coefficients is sufficient to reconstruct the full value function and distribution, we will also sometimes refer to ω t as the reduced value function and to β t as the reduced distribution. Following the same steps as above, we thus reduce our original system (LINEAR) to E t dω t S T B vv S S T B vp B pg X S T B vp B pz dβ t = X T B gv S X T (B gg +B gp B pg )X X T B gp B pz dz t ν ω t β t Z t dt. We show in Appendix A.2 that the state reduction of the reduced linear model is guaranteed to be accurate if we increase the order of the observability matrix k g until convergence of the impulse response functions of the p t. Because of linearity, the convergence of these impulse responses implies that we have computed the subspace of ĝ t necessary for accurately predicting the dynamics of p t. The dynamics of p t are in turn sufficient to compute households decisions. Hence, to check whether our state reduction step is accurate, we simply need to vary k g until the impulse responses of p t do not change. Similarly, to check whether the spline approximation of the value function v t is accurate, we increase the number of nodes in the approximation until the fit of the spline no longer improves. 2.4 Performance of the Method in Krusell-Smith Model The Krusell-Smith model is a useful environment for evaluating our model reduction methodology because it is possible to solve the full unreduced model as a benchmark. For this illustration we parameterize the model following Den Haan, Judd and Julliard (21). A unit of time is one quarter. We set the rate of time preference ρ =.1 and the coefficient of relative risk aversion γ = 2. Capital depreciates δ =.25 per quarter and the capital share is α = 1. We choose the idiosyncratic shock process following Den Haan, Judd and Julliard 3 (21), assuming the distribution of shocks is constant. We set the aggregate shock process to match an approximate quarterly persistence Corr(Z t+1,z t ) = e ν 1 ν =.95 and innovation volatility σ =.7. Finally, the individual asset grid ranges from a 1 = to a 1 = 1 with I = 1 asset grid points. Hence the total number of grid points is N = 2 and the size of the dynamical system is 4. We are able to substantially reduce the size of the model without sacrificing accuracy; setting the order of the observability matrix k = 1 and using k v = 12 spline knots in the 18

19 wealth dimension provides an extremely accurate approximation of the model s dynamics. Figure 1 shows that the impulse responses of key aggregate variables in the reduced model are almost exactly identical to the full, unreduced model, despite bringing the model from a 4-dimensional to a 3-dimensional dynamical system. 17 These results are consistent with Krusell and Smith (1998) s finding of approximate aggregation in the discrete time version of this model, using a computationally distinct procedure and accuracy measure. Figure 2 illustrates this visually by plotting the elements of the basis X = [x 1,...,x k ] up to four orders in the Taylor expansion. 18 For example, the first subfigure plotssplits the 2-dimensional vector x 1 into the two 1-dimensional vectors corresponding to each of the two productivity types and plots each as a function of the asset grid. Ascanbeseeninthefigure, bothvectorsequaltheassetgrida = [a 1,...,a I ] T, andhence the first basis vector equals x 1 = [ a a]. Recall from (11) that the reduced distribution is given by the k g -dimensional vector β t = X T g t. Thus, the first element of the reduced distribution β 1t = x T 1g t = [ a a] T g t which is exactly the mean of the distribution, i.e. the aggregate capital stock. This is intuitive because the mean is, of course, necessary to compute aggregate capital and therefore prices. The more interesting result is that all the higher-order elements ofx, i.e. x 2,x 3 andso on, quickly converge to near-zeroconstants, i.e. theydo not addmuch additional information to the approximation. Thus, in this simple Krusell-Smith economy, our model-free model reduction method in fact confirms Krusell and Smith s approximate aggregation result. With or without dimensionality reduction, our method solves and simulates the model in less than half a second. Table 1 reports the running time of using our Matlab code suite on a desktop PC. Although reduction is not necessary to solve this simple model, it nevertheless reduces running time by nearly 6%. In the two-asset model in Section 3, model reduction is necessary to even solve the model. 2.5 Matlab Code Package If a researcher can compute the steady state of their heterogeneous agent model, our Matlab code package computes the aggregate dynamics essentially for free. The user simply provides two inputs: the steady state values of the model s variables and a function which defines the equilibrium conditions. Our package then provides the tools to compute the first-order Taylor expansion of the equilibrium conditions, using our automatic differentiation software which 17 The impulse responses fully characterize the model s dynamics because the solution is linear. 18 Technically, our basis vector are ortho-normalized during the Arnoldi iteration; for interpretability, we have not normalized them here. 19

20 Figure 1: Impulse Responses to TFP Shock in Krusell-Smith Model % deviation from s.s TFP Full model Reduced model Output % deviation from s.s Consumption Quarters 1.5 Investment Quarters Notes: We simulate the model by discretizing the time dimension with step size dt =.1. We define an impulse response as the response of the economy to a one standard deviation innovation to the TFP process over an instant of time dt. Full model refers to model solved without model reduction and reduced model with reduction, using k g = 1 and k v = 12. 2

21 Figure 2: Basis Vectors in Distribution Reduction 15 1 Low productivity High productivity Notes: The first four elements of the basis X, computed without ortho-normalization in the Arnoldi iteration. 21

22 Table 1: Run Time for Solving Krusell-Smith Model Full Model Reduced Model Steady State.88 sec.75 sec Derivatives.28 sec.27 sec Dim reduction Linear system.138 sec Simulate IRF.51 sec.13 sec.5 sec.8 sec Total.35 sec.128 sec Notes: Time to solve Krusell-Smith model once on Dell Precision T581 with 3.1 GHz processor and 32 GB RAM, using Matlab code suite. Full model refers to solving model without model reduction and reduced model with reduction, using k g = 1 and k v = 12. Steady state reports time to compute steady state. Derivatives reports time to compute derivatives of discretized equilibrium conditions. Dim reduction reports time to compute both the distribution and value function reduction. Linear system reports time to solve system of linear differential equations. Simulate IRF reports time to simulate impulse responses reported in Figure 1. Total is the sum of all these tasks. exploits sparsity; reduces the distribution following Sections and 2.3.2; reduces the value function using non-uniformly spaced quadratic splines following Section 2.3.3; solves the resulting system of linear differential equations using a solver that exploits sparsity; and simulates the model to produce impulse responses or time series statistics. 2.6 Micro Heterogeneity and Macro Nonlinearities A common motivation for studying heterogeneous agent macro models is that movements in the distribution of agents can potentially generate nonlinear dynamics in aggregate variables. For example, researchers may be interested in how the response of the economy to a shock depends on the initial distribution of MPCs across households. Despite relying on linear approximations, our method can preserve these types of nonlinearities. To fix ideas, consider for example the impulse response of aggregate consumption C t to a productivity shock Z t, starting from some initial distribution g (a,z). We first compute the impact effect of the shock using the linear approximation of the value function v (a,z) = v(a,z)+ v (a,z) from (SOLUTION) to back out the individual consumption decision using the first order condition c (a,z) = ( a v (a,z)) 1 γ. Integrate this over households 22

23 to get aggregate consumption C = c (a,z)g (a,z)dadz. Clearly, this impact response will depend on the initial distribution g (a,z), allowing the effect of the shock to be state-dependent. 19 To compute the dynamics after impact, we update the distribution using the fully nonlinear KFE in (NON-LINEAR) instead of the linearized version in (LINEAR): dg dt = A(v ;p ) T g This procedure allows us to preserve size-dependence because larger shocks potentially induce non-proportional movements in the individual state space, and therefore different distributional dynamics going forward. The only instant at which this is not true is the impact period t = because at t = the value function is itself linear in the aggregate shock (see footnote 19). However, the impact period is arbitrarily small in continuous time. After the impact period we repeat this procedure throughout the course of the simulation. 2 3 Aggregate Consumption Dynamics Although the Krusell-Smith model was a useful environment to explain our computational method, it is too simple to illustrate the advantages of our method over others. We now turn to using our method to solve a quantitatively realistic two-asset model with aggregate 19 To see this even more directly, note that one can write the deviation of initial consumption from steady state in the form ĉ (a,z) D cz (a,z)z where the coefficients D cz (a,z) are found as follows: ĉ (a,z) = ( a v(a,z)) 1 γ 1 a v (a,z) = ( a v(a,z)) 1 γ 1 a D vz (a,z)z := D cz (a,z)z. and where D vz (a,z) are the elements of D vz in (SOLUTION), i.e. the different derivatives for each point in the state space. Therefore the aggregate derivative depends on the distribution over these points, i.e. state dependence: Ĉ D cz (a,z)g (a,z)dadz Z. At the same time, the initial impulse response Ĉ does not feature size dependence with respect to the initial impulse Z which enters linearly. 2 Again, a special feature of the Krusell and Smith model is that prices are directly defined by the distribution, but in more general models prices may only be defined implicitly through market clearing conditions. In those cases we use the linearized pricing equation from the linearized solution of the model. Although this imposes that prices are a linear function of the distribution, it still allows for nonlinear dynamics of the prices because the distribution evolves fully nonlinearly. 23

24 shocks, which was previously intractable. 3.1 Model The household side of the model closely follows Kaplan, Moll and Violante (216) so we keep our exposition of it brief. The firm side follows the standard real business cycle model with persistent shocks to productivity growth Environment Households There is a unit mass of households indexed by j [,1]. At each instant of time households hold liquid assets b jt, illiquid assets a jt, and have labor productivity z jt. Households die with an exogenous Poisson intensity ζ and upon death give birth to an offspring with zero wealth a jt = b jt = and labor productivity drawn from its ergodic distribution. There are perfect annuity markets, implying that the wealth of deceased households is distributed to other households in proportion to their asset holdings. Each household has preferences over consumption c jt represented by the expected utility function E c1 γ (ρ+ζ)t jt e 1 γ. Ahousehold withlaborproductivityz jt earnslaborincomew t z jt andpaysalinearincome tax at rate τ. Each household also receives a lump-sum transfer from the government T Z t, where Z t is aggregate productivity, described below. Labor productivity is a discrete process z jt {z 1,...,z J } which we calibrate below and households switch from state z to state z with Poisson intensity λ zz. The liquid asset b jt pays a rate of return r b t. Households can borrow in liquid assets up to an exogenous limit b Z t. The interest rate on borrowing is r b t = rt b +κ where κ > is a wedge between borrowing and lending rates. Let rt(b b t ) denote the interest rate function taking both of these cases into account. The illiquid asset a jt pays a rate of return rt a and cannot be borrowed. It is illiquid in the sense that there is a cost χ(d jt,a jt ) Z t of depositing d jt into or out of the illiquid account. The transaction cost function is given by d jt χ(d jt,a jt ) = χ d jt +χ 1 a χ2 jt. a jt 24